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The cohomology groups have a richer, or at least more familiar, algebraic structure than the homology groups. Firstly, they form a differential graded algebra as follows: the graded set of groups form a graded R-module; this can be given the structure of a graded R-algebra using the cup product; the Bockstein homomorphism β gives a differential.
Hatcher, Allen (2001), Algebraic Topology, Cambridge University Press, ISBN 0-521-79540-0, MR 1867354 "Cohomology" , Encyclopedia of Mathematics , EMS Press , 2001 [1994] . May, J. Peter (1999), A Concise Course in Algebraic Topology (PDF) , University of Chicago Press , ISBN 0-226-51182-0 , MR 1702278
Allen Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002. ISBN 0-521-79540-0. A modern, geometrically flavored introduction to algebraic topology. The book is available free in PDF and PostScript formats on the author's homepage. Kainen, P. C. (1971). "Weak Adjoint Functors". Mathematische Zeitschrift. 122: 1– 9.
Hatcher, A., Algebraic Topology, Cambridge University Press (2002) ISBN 0-521-79540-0. Detailed discussion of homology theories for simplicial complexes and manifolds, singular homology, etc. May JP (1999). A Concise Course in Algebraic Topology (PDF). University of Chicago Press. Archived (PDF) from the original on 2022-10-09
An H-space consists of a topological space X, together with an element e of X and a continuous map μ : X × X → X, such that μ(e, e) = e and the maps x ↦ μ(x, e) and x ↦ μ(e, x) are both homotopic to the identity map through maps sending e to e. [2]
In algebraic topology, the pushforward of a continuous function : between two topological spaces is a homomorphism: () between the homology groups for . Homology is a functor which converts a topological space X {\displaystyle X} into a sequence of homology groups H n ( X ) {\displaystyle H_{n}\left(X\right)} .
In mathematics, more specifically algebraic topology, a pair (,) is shorthand for an inclusion of topological spaces:.Sometimes is assumed to be a cofibration.A morphism from (,) to (′, ′) is given by two maps : ′ and : ′ such that ′ =.
Allen Hatcher and William Thurston, A presentation for the mapping class group of a closed orientable surface, Topology 19 (1980), no. 3, 221–237. Allen Hatcher, On the boundary curves of incompressible surfaces, Pacific Journal of Mathematics 99 (1982), no. 2, 373–377.